Molecular Motors: The movement on a microtubule of a single two-headed kinesin motor carrying a cargo, as investigated experimentally in some in vitro motility assays, was studied theoretically. The purpose is to study how the Brownian motion of the cargo affects the turnover rate of the motor. We developed a theoretical formalism useful for this purpose. Using a simple two-state cycle model for the motor, we found that the velocity of movement of the motor-cargo complex on microtubule increases when the diffusion coefficient of the cargo increases or the stiffness of the elastic element connecting the cargo to the motor is reduced. In other words, the motor moves faster when the Brownian motion of the cargo increases. This implies that the effect of Brownian motion of the cargo cannot be neglected and the present formalism should be used when analyzing the motility data obtained from in vitro motility assays. (Publication: Biophysical Chemistry 91: 79-91 (2001)). (Chen, Yan, and Rubin). Actomyosin Regulation: Binding kinetics of myosin sub-fragment 1 (S1) to regulated actin in the presence and absence of calcium were studied theoretically for two kinds of cooperative ligand-binding models. In one model (the Hill model), each tropomyosin-troponin-actin complex can exist in two states with different S1 binding affinities and different neighbor-neighbor cooperativities. In the other model (the Geeves model), the complex can exist in three states with state transitions being dependent on the "state" of the S1. The two models are quite different in the mechanisms of regulating the activation of muscle contraction. It has been argued qualitatively before that the Hill model could not account for some of the S1 binding kinetics. We showed in this study that this is not the case. We showed that both models were indistinguishable in their ability/inability to account for existing kinetic and equilibrium binding data. (Publications: Biophysical Journal 80: 2338-2349 (2001); 82: 1679-1681 (2002)) (Chen, Yan, Chalovich, and Brenner). Caldesmon Inhibition of Myosin Binding to Actin: Our ultimate goal of this project is to study the regulation of smooth muscle contraction by the actin-binding protein, caldesmon. Specifically, we want to model the observed inhibition of caldesmon on S1 ATPase activity. Toward this goal, it is useful is to first study the effect of caldesmon on the binding of S1 to actin. It is known that a caldesmon molecule can cover seven actin monomers while an S1 molecule can cover only one. Earlier, we have shown that the equilibrium binding of S1 and caldesmon to actin was better described by the ?mosaic multiple? binding model, in which S1 can bind not only to bare actin sites, but also to those already covered by a caldesmon, and vice versa. In this study, we showed that this conclusion was not quite correct based on the kinetic curves of S1 binding as a function of caldesmon. Instead, the kinetics of S1 binding in the presence of caldesmon is better described as pure competitive. (Publication: Biochemistry 40: 5757-5764 (2001) (Chen, Yan, Chalovich). Brownian Ratchets: A charged Brownian particle placed in a linear periodic saw-tooth (asymmetric ratchet) potential can be made to move in one direction if the potential is made to flash on-and-off randomly. In this study we examined the relation between the direction of movement and the asymmetry of the potential. We found that by kinking one arm of the ratchet the direction of movement of the particle could be reversed depending on the frequency of the fluctuation and the location and the degree of the kink. Results obtained in this study are very useful in designing flashing-potential apparatuses for the separation of proteins based on their sizes and charges and the viscosity of the medium. (Publication: Journal of Theoretical Biology 210: 141-150 (2001)) (Chen, Yan). Microtubule Dynamics and Related Problems: The dynamic properties of microtubule polymerization have been studied using both the discrete and the continuum models. To study the differences between the two models, the first moment and the higher cumulant moments of the tip distribution function up to the 6th have been calculated for the two models using Mathematica. We show that the two methods are approximately equivalent when experimentally obtained kinetic parameters are used. A discrete one-dimensional random walk with absorption on one end has been used to derive the first passage time for cycle completion of a two-state cycle diagram. This first passage time is then used to evaluate the mean and the variance of cycle completions at any given time. The result can be used to discuss the ?randomness? of the velocity of kinesin movement on microtubule measured in some motility assays. (Rubin, Chen).